I.6: Nonsingular Curves
6.1. #to_work
Recall that a curve is rational if it is birationally equivalent to \(\mathbf{P}^1(\mathrm{Ex} .4 .4)\). Let \(Y\) be a nonsingular rational curve which is not isomorphic to \(\mathbf{P}^1\).
- Show that \(Y\) is isomorphic to an open subset of \(\mathbf{A}^1\).
- Show that \(Y\) is affine.
- Show that \(A(Y)\) is a unique factorization domain.
6.2. An Elliptic Curve. #to_work
Let \(Y\) be the curve \(y^2=x^3-x\) in \(\mathbf{A}^2\), and assume that the characteristic of the base field \(k\) is \(\neq 2\). In this exercise we will show that \(Y\) is not a rational curve, and hence \(K(Y)\) is not a pure transcendental extension of \(k\).
- Show that \(Y\) is nonsingular, and deduce that \(A=A(Y) \simeq k[x, y] /\left(y^2-x^3+x\right)\) is an integrally closed domain.
- Let \(k[x]\) be the subring of \(K=K(Y)\) generated by the image of \(x\) in \(A\). Show that \(k[x]\) is a polynomial ring, and that \(A\) is the integral closure of \(k[x]\) in \(K\).
- Show that there is an automorphism \(\sigma: A \rightarrow A\) which sends \(y\) to \(-y\) and leaves \(x\) fixed. For any \(a \in A\), define the norm of \(a\) to be \(N(a)=a \cdot \sigma(a)\). Show that \(N(a) \in k[x], N(1)=1\), and \(N(a b)=N(a) \cdot N(b)\) for any \(a, b \in A\).
- Using the norm, show that the units in \(A\) are precisely the nonzero elements of \(k\). Show that \(x\) and \(y\) are irreducible elements of \(A\). Show that \(A\) is not a unique factorization domain.
- Prove that \(Y\) is not a rational curve (Ex. 6.1). See (II, 8.20.3) and (III, Ex. 5.3) for other proofs of this important result.
6.3. #to_work
Show by example that the result of \((6.8)\) is false if either (a) \(\operatorname{dim} X \geqslant 2\), or (b) \(Y\) is not projective.
6.4. #to_work
Let \(Y\) be a nonsingular projective curve. Show that every nonconstant rational function \(f\) on \(Y\) defines a surjective morphism \(\varphi: Y \rightarrow \mathbf{P}^1\), and that for every \(P \in \mathbf{P}^1\), \(\varphi^{-1}(P)\) is a finite set of points.
6.5. #to_work
Let \(X\) be a nonsingular projective curve. Suppose that \(X\) is a (locally closed) subvariety of a variety \(Y\) (Ex. 3.10). Show that \(X\) is in fact a closed subset of \(Y\). See (II, Ex. 4.4) for generalization.
6.6. Automorphisms of \(\mathbf{P}^1\). #to_work
Think of \(\mathbf{P}^1\) as \(\mathbf{A}^1 \cup\left\{{\infty}\right\}\). Then we define a fractional linear transformation of \(\mathbf{P}^1\) by sending \(x \mapsto(a x+b)/(c x+d)\), for \(a, b, c, d \in k\), and \(ad-b c \neq 0\).
- Show that a fractional linear transformation induces an automorphism of \(\mathbf{P}^1\) (i.e., an isomorphism of \(\mathbf{P}^1\) with itself). We denote the group of all these fractional linear transformations by \(\operatorname{PGL}(1)\).
- Let Aut \(\mathbf{P}^1\) denote the group of all automorphisms of \(\mathbf{P}^1\). Show that Aut \(\mathbf{P}^1 \simeq\) Aut \(k(x)\), the group of \(k\)-automorphisms of the field \(k(x)\).
- Now show that every automorphism of \(k(x)\) is a fractional linear transformation, and deduce that \(\mathrm{PGL}(1) \rightarrow\) Aut \(\mathbf{P}^1\) is an isomorphism.
Note: We will see later (II. 7.1.1) that a similar result holds for \(\mathbf{P}^n\) : every automorphism is given by a linear transformation of the homogeneous coordinates.
6.7. #to_work
Let \(P_1, \ldots, P_r, Q_1, \ldots, Q_s\) be distinct points of \(\mathbf{A}^1\). If \(\mathbf{A}^1-\left\{P_1, \ldots, P_r\right\}\) is isomorphic to \(\mathbf{A}^1-\left\{Q_1, \ldots, Q_{\diamond}\right\}\), show that \(r=s\). Is the converse true? Cf. (Ex. 3.1).